Nonuniform mixing

Fluid mixing usually involves the interplay between advection and diffusion, which together cause any initial distribution of passive scalar to homogenize ultimately reach a uniform state. However, this scenario only holds when velocity field is nondivergent has no normal component boundary. If either condition unmet, such as for active particles in bounded region, floating particles, or filters, then ultimate state after long time not uniform, may be dependent. We show that those cases nonuniform it preferable characterize degree terms an f-divergence, generalization relative entropy, use $L^1$ norm. Unlike concentration variance ($L^2$ norm), f-divergence norm always decay monotonically, even mixing, facilitates measuring rate mixing. by example flows mix well case can drastically different from efficient uniformly flows.